Question

if x=9-4â5,then find the value of x^2+1/x^2

Answer 1

${x}^{2} + \frac{1}{ {x}^{2} } \: = \: ? \: \\ \\ x = 9 - 4 \sqrt{5} \\ x = {(2 - \sqrt{5} )}^{2} \\ \\ \frac{1}{x} = \frac{1}{9 - 4 \sqrt{5} } \times \frac{9 + 4 \sqrt{5} }{9 + 4 \sqrt{5} } = 9 + 4 \sqrt{5} \\ \\ \frac{1}{x} = {(2 - \sqrt{5} )}^{2} \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = ({{x + \frac{1}{x}})}^{2} - 2(x)( \frac{1}{x} ) \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = {18}^{2} \: - \: 2 \: \\ \\ \\ \bf \: \large \boxed{{x}^{2} + \frac{1}{ {x}^{2} } = 322}$